Let \(\xi \)  be a measurable partition of \(X\) . Note that, for \(\mu \) -typical \(x\in X\)  at which the conditional measure \(\mu ^{\xi }_{x}\)  is well-defined, the fibre measure space \((\xi (x),\mathcal {B}|_{\xi (x)}, \mu ^{\xi }_{x})\)  is also a Lebesgue space. Therefore we may apply Rokhlin's conditional measure theorem to \(\mu ^{\xi }_{x}\)  with respect to the measurable partition \(\eta =\lbrace \phi ^{-1}(y):y\in E\rbrace \)  restricted to \(\xi (x)\) . Let \( \widehat{\xi }\vee \widehat{\eta }\)  denote the smallest sub \(\sigma \) -algebra containing both \(\widehat{\xi }\)  and \(\widehat{\eta }\) , which is equal to \(\widehat{\xi \vee \eta }\) . We have the following extension of [1]}.
